Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C27 · Schur multiplier (H2) = C25 |
| direct product | C27×#578 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C27 | C22×D4 |
| commutator | C4 | C24×C27 |
| Frattini, cosocle | C4×C9 | C24×C3 |
| socle | C2×C3 | C22×D4×C9 |
| prime radical | C3×#578 | C9 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C27 ◁ #41×C27 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,33,37)(30,38,41,51)(31,35,42,49)(32,44,46,36)(34,45,48,40)(39,52,53,59)(43,56,55,50)(47,57,58,54);(28,30,33,41)(29,34,37,48)(31,39,42,53)(32,45,46,40)(35,47,49,58)(36,51,44,38)(43,57,55,54)(50,59,56,52);(28,31)(29,35)(30,39)(32,43)(33,42)(34,47)(36,50)(37,49)(38,52)(40,54)(41,53)(44,56)(45,57)(46,55)(48,58)(51,59);(31,42)(35,49)(39,53)(43,55)(47,58)(50,56)(52,59)(54,57)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2709);